Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences

Yogesh Kumar Gupta, Mamta Singh, Omprakash Sikhwal

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Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences

Yogesh Kumar Gupta1,, Mamta Singh2, Omprakash Sikhwal3

1Schools of Studies in Mathematics, Vikram University Ujjain, (M. P.) India

2Department of Mathematical Sciences and Computer application, Bundelkhand University, Jhansi (U. P.)

3Department of Mathematics, Mandsaur Institute of Technology, Mandsaur (M. P.) India

Abstract

The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula Fn=Fn-1+Fn-2, , and F0=0, F1=1, where Fn is a nth number of sequence. Many authors have been defined Fibonacci pattern based sequences which are popularized and known as Fibonacci-Like sequences. In this paper, Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation Bn=Bn-1+Bn-2, with B0=2s, B1=s+1, where s being a fixed integers. Some identities of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences are presented by Binet’s formula. Also some determinant identities are discussed.

Cite this article:

  • Gupta, Yogesh Kumar, Mamta Singh, and Omprakash Sikhwal. "Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences." Turkish Journal of Analysis and Number Theory 2.6 (2014): 233-238.
  • Gupta, Y. K. , Singh, M. , & Sikhwal, O. (2014). Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences. Turkish Journal of Analysis and Number Theory, 2(6), 233-238.
  • Gupta, Yogesh Kumar, Mamta Singh, and Omprakash Sikhwal. "Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences." Turkish Journal of Analysis and Number Theory 2, no. 6 (2014): 233-238.

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1. Introduction

The Fibonacci and Lucas sequences are well-known examples of second order recurrence sequences. The Fibonacci numbers are perhaps most famous for appearing in the rabbit breeding problem, introduced by Leonardo de Pisa in 1202 in his book called Liber Abaci. As illustrate in the tome by Koshy [15] the Fibonacci and Lucas number are arguable two of the most interesting sequence in all of mathematics. Many identities have been documented in an extensive list that appears in the work of Vajda [14], where they are proved by algebra means, even though combinatorial proof of many of these interesting identities. We introduced Generalized Fibonacci-Like Sequence and some identities Fibonacci numbers, Lucas number's and their generalization have many interesting Properties and application to almost every field.

The Fibonacci sequence [5] is a sequence of numbers starting with integer 0 and 1, where each next term of the sequence calculated as the sum of the previous two.

(1.1)

The similar interpretation also exists for Lucas sequence. Lucas sequence [10] is defined by the recurrence relation,

(1.2)

In this paper, we present various properties of the Generalized Fibonacci-Like sequence (GFLS) associated with Fibonacci and Lucas sequences {Bn} defined by

(1.3)

The Binet's formula for Fibonacci sequence is given by

(1.4)

where Golden ratio = 1.618

and Golden ratio = -0.618

Similarly, the Binet's formula for Lucas sequence is given by

2. Preliminary Results Generalized Fibonacci-Like Sequence

We need to introduce some basic results of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences {Bn} is defined by recurrence relation:

(2.1)

With initial conditions B0 = 2s and B1 = s+1.

The associated initial Condition B0 and B1 are the sum of initial condition of generalized Fibonacci-Like sequence respectively.

(2.2)

The few terms of above sequence are 2s, s+1, 1+3s, 2+4s, 3+7s, and so on.

The relation between Fibonacci sequence and Generalized Fibonacci-Like Sequence can be written as

The recurrence relation (1.1) has the characteristic equation which has two roots

Now notice a few things about and

Using these two roots, we obtain Binet’s recurrence relation

3. Generating Function

Now we state derive generating function of generalized Fibonacci-Like sequence

(3.1)

Let's apply power series to sequence { Bn }

Let

Where Bn is nth term of sequence .

This is called generating series of Generalized Fibonacci - Like Sequence { Bn }.

Now multiplying the generating series

Therefore,

Hence

4. Properties of Generalized Fibonacci- Like Sequence

Despite its simple appearance the Generalized Fibonacci-Like sequence {Bn} contains a wealth of subtle and fascinating properties [4, 6, 9, 12].

Sum of n First terms:

Theorem (4.1). Let Bn be the nth Fibonacci-Like number, then Sum of the first n terms of generalized Fibonacci-Like sequence is

(4.1)

Proof: we know that the follows relation holds:

Term wise addition of all above equations, we obtain

Sum of First n terms with even indices

Theorem (4.2). Let Bn be the nth Fibonacci-Like sequence, then Sum of the first n terms with even indices is

(4.2)

Sum of First n terms with square indices:

Theorem (4.3). Let Bn be the nth Fibonacci-Like sequence, then Sum of the square of first n terms is

(4.3)

Sum of First n terms with odd indices:

Theorem (4.4). Let Bn be the nth Fibonacci-Like sequence, then Sum the first n terms with odd indices is

(4.4)

Now we state and prove some nice identities similar to those obtained for Fibonacci and Lucas sequences [1, 2, 4, 12].

5. Some Identities Generalized Fibonacci- Like Sequence

In this section, some identities of Generalized Fibonacci-Like sequence are presented which can be easily derived by Explicit sum formula using generating function and Binet’s formula. Authors [5, 6] have been described such type identities.

Explicit Sum Formula:

Theorem (5.1). The explicit sum formula for Generalized Fibonacci-Like sequence is given by For positive integer n, Prove that

(5.1)

Proof: By equation (2.1), it follows that

Hence B2n = .

Theorem (2). The explicit sum formula for Generalized Fibonacci-Like sequence is given by For positive integer n,

(5.2)

Theorem (5.3). For every positive integer n, prove that

(5.3)

Proof: Let n be fixed and we Proved by inducting on m.

When m = 0, then

Which is true.

When m=1, then

which also is true.

Now assume that identity is true for m = k+1, then by assumption

(5.4)
(5.5)

Adding equation (5.4) and (5.5), we get

Which is precisely our identity when k = m

Hence

Theorem (5.4). For every positive integer n, prove that

(5.6)

Proof: we shall have proved this identity by induction matched over n.

which is also true for n=0.

When n = 1 than

which is also true for n = 1.

For n = k

For n = k which is also true.

Now assume that identity is true for n =1, 2, 3.........k and

We so that it holds:

For n = k+1, then by assumption

Which is also true, for n = k+1

Hence, the result is true for all.

Theorem (5.5). For every positive integer n, prove that

(5.7)

Proof: we shall Prove this identity by induction over n, for n= 2

Now suppose that identity hold for n =k-1, n = k-2 than,

(5.8)
(5.9)

On adding equation (5.8) & (5.9) we get,

which is true for n = k,

Theorem(5. 6). For every positive integer n,

(5.10)

Proof. By using Binet’s formula, we have

This is completes the proof.

Theorem (5.7). For every positive integer

(5.11)

Proof. By using Binet’s formula, we have

This is completes the proof.

Theorem (5.8). For positive integer n, prove that

(5.12)

This can be derived same as theorem (1.4)

Theorem (5.9). For positive integer n, prove that

(5.13)

This can be derived same as theorem (1.4).

Theorem (5.10). For every integer , prove that

(5.14)

This can be derived same as theorem (1.4)

Theorem (5.11). For every integer , prove that

(5.15)

This can be derived same as theorem (1.4).

Theorem (5.12). For every integer , prove that

(5.16)

This can be derived same as theorem (1.4).

6. Connection Formulae

In this section, connection formulae of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences, induction method are presented.

Theorem (6.1). For positive integer n, Prove that

(6.1)

Proof: We shall prove this identity by induction. It is easy to show that for n = 3

Now suppose the identity holds n = k-1, n = k-2. Then,

(6.2)
(6.3)

On adding equation (6.2) and (6.3), we get

Which is precisely our identity when n = k.

Hence 2s Fn-1 = Bn-1 - Bn-2, n .

Theorem (6.2). For positive integer n, Prove that

(6.4)

Proof: We shall Prove this identity by induction over n. for n = 2

Now suppose the identity holds for n = k-1, n = k-2. Then,

(6.5)
(6.6)

Adding equation (6.5) and (6.6), we get

Which is true for n = k,

Hence 2s Ln-1 = Bn - Bn-1, n .

Theorem (6.3). For positive integer n, prove that

(6.7)

Theorem (6.4). For positive integer n, prove that

(6.8)

Theorem (6.5). For positive integer n, prove that

(6.9)

Theorem (6.6). For positive integer n, prove that

(6.10)

7. Some Determinant Identities

There is a long tradition of using matrices and determinants to study Fibonacci numbers. Problems on determinants of Fibonacci sequence and Lucas sequence are appeared in various issues of Fibonacci Quarterly. T. Koshy [15] explained two chapters on the use of matrices and determinants. Many determinant identities of generalized Fibonacci sequence are discussed in [4, 6] and [11]. In this section some determinant identities of Generalized Fibonacci-Like sequence are presented. Entries of determinants are satisfying the recurrence relation of Generalized Fibonacci-Like sequence and other sequences.

Theorem (7.1). Let n be a positive integer. Then

Proof: Let

(7.1)

And

(7.2)
(7.3)

Now substituting the value of equation (7.2) & (7.3) in (7.1), we get

Applying

Applying R2

(7.4)

Again substituting the values of the equation (7.2) and (7.3) in (7.4).

We get

Hence

Similarly we can derive following identities:

Theorem (7.2). For every integer n ≥ 2, prove that

(7.5)

Theorem (7.3). For any integer n ≥0, prove that

(7.6)

Theorem (7.4). For every positive integer n, prove that

(7.7)

Theorem (7.5). For every positive integer n, prove that

(7.8)

The identities from (7.1) to (7.4) can be proved similarly.

8. Conclusions

In this paper, Generalized Fibonacci-Like sequence is introduced. Some standard identities of generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences have been obtained and derived using Binet’s formula. Also some determinant identities have been established and derived.

Acknowledgement

We would like to thank the anonymous referees for numerous helpful suggestions.

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